An arbitrary order variationally consistent integration for Galerkin meshfree methods

Because most approximation functions employed in meshfree methods are rational functions with overlapping supports, sufficiently accurate domain integration becomes costly, whereas insufficient accuracy in the domain integration leads to suboptimal convergence. In this paper, we show that it is possible to achieve optimal convergence by enforcing variational consistency between the domain integration and the test functions, and optimal convergence can be achieved with much less computational cost than using higher-order quadrature rules. In fact, stabilized conforming nodal integration is variationally consistent, whereas Gauss integration and nodal integration are not. In this work the consistency conditions for arbitrary order exactness in the Galerkin approximation are set forth explicitly. The test functions are then constructed to be variationally consistent with the integration scheme up to a desired order. Attempts are also made to correct methods that are variationally inconsistent via modification of test functions, and several variationally consistent methods are derived under a unified framework. It is demonstrated that the solution errors of PDEs due to quadrature inaccuracy can be significantly reduced when the variationally inconsistent methods are corrected with the proposed method, and consequently the optimal convergence rate can be either partially or fully restored.